Gautschi's Inequality for Gamma function

Prove that:

$$x^{1-s} < \frac{\Gamma(x+1)}{\Gamma(x+s)} < (x+1)^{1-s},\qquad x > 0,\; 0 < s < 1$$

which is better known as Gautschi's Inequality , due to Walter Gautschi.

Solution

 The strict log-convexity of $\Gamma$ (see here)  implies that for $0< s <1$,

$$\Gamma(x+s)<\Gamma(x)^{1-s}\Gamma(x+1)^s=x^{s-1}\Gamma(x+1)$$

which yields

$$x^{1-s}<\frac{\Gamma(x+1)}{\Gamma(x+s)}\tag{1}$$

Again by the strict log-convexity of $\Gamma$,

$$\Gamma(x+1)<\Gamma(x+s)^s\Gamma(x+s+1)^{1-s}=(x+s)^{1-s}\Gamma(x+s) $$
which yields

$$\frac{\Gamma(x+1)}{\Gamma(x+s)}<(x+s)^{1-s}<(x+1)^{1-s}\tag{2}$$

Combining $(1)$ and $(2)$ yields

$$x^{1-s}<\frac{\Gamma(x+1)}{\Gamma(x+s)}<(x+1)^{1-s}$$

and the inequality is proved.